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Kite's Genchain mode numbering: Difference between revisions

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m TallKite moved page Naming Rank-2 Scales using Mode Numbers to Kite's Method of Naming Rank-2 Scales using Mode Numbers: Avoid implying it's the only way to name rank-2 scales
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The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below in "Rationale"). '''<u>Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]</u>.''' There are several disadvantages of only using chroma-positive generators. See the critique of UDP at the end of this article.
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below in "Rationale"). '''<u>Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]</u>.''' There are several disadvantages of only using chroma-positive generators. See the critique of UDP in the "Rationale" section below.


Pentatonic meantone scales:
Pentatonic meantone scales:
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'''[[Porcupine]] aka Triyo''' has a [[pergen]] of (P8, P4/3) and a generator of vM2 = ~10/9. Porcupine[7] modes, using [[Ups and Downs Notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.
'''[[Porcupine]] aka Triyo''' has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2. The enharmonic interval is v<sup>3</sup>A1. Because the generator is a 2nd, the genchain resembles the scale. Porcupine[7] modes, using [[Ups and Downs Notation|ups and downs notation]]:


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| | D Ev F^ G Av Bb^ <u>'''C'''</u>
| | D Ev F^ G Av Bb^ <u>'''C'''</u>
|}
|}
'''[[Sensi]] aka Sepgu''' has pergen (P8, WWP5/7). The ~9/7 generator is a ^<sup>3</sup>d4 = v<sup>4</sup>A3, and the enharmonic is a ^<sup>7</sup>dd2. Sensi[8] modes:
'''[[Sensi]] aka Sepgu''' has pergen (P8, WWP5/7). The ~9/7 generator is both a ^<sup>3</sup>d4 and a v<sup>4</sup>A3, and the enharmonic interval is ^<sup>7</sup>dd2. Sensi[8] modes:
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A MODMOS scale can have alternate names. The ascending melodic minor scale could also be called 2nd Meantone[7] b3 (major scale with a minor 3rd), or as 4th Meantone[7] #7 (dorian with a major 7th).  
A MODMOS scale can have alternate names. The ascending melodic minor scale could also be called 2nd Meantone[7] b3 (major scale with a minor 3rd), or as 4th Meantone[7] #7 (dorian with a major 7th).  


'''Meantone''' MODMOS scales, with alternative names included only if they don't have more alterations than the original:
'''Meantone''' MODMOS scales, with alternative names in italics and parentheses. Alternatives that have more alterations than the original aren't listed:


{| class="wikitable"
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| | C * D <u>'''A'''</u> E B F# * G#
| | C * D <u>'''A'''</u> E B F# * G#
|-
|-
| style="text-align:center;" | (Major with b3)
| style="text-align:center;" | ''(Major with b3)''
| | 2nd Meantone[7] b3
| | ''(2nd Meantone[7] b3)''
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
|-
|-
| style="text-align:center;" | (Dorian with #7)
| style="text-align:center;" | ''(Dorian with #7)''
| | 4th Meantone[7] #7
| | ''(4th Meantone[7] #7)''
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
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| | F C * * <u>'''A'''</u> E B * * G# D#
| | F C * * <u>'''A'''</u> E B * * G# D#
|-
|-
| style="text-align:center;" | (Lydian with b3 b6)
| style="text-align:center;" | ''(Lydian with b3 b6)''
| | 1st Meantone[7] b3 b6
| | ''(1st Meantone[7] b3 b6)''
|"
|"
| style="text-align:center;" | "
| style="text-align:center;" | "
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| | Bb F * * D <u>'''A'''</u> E * * C# G#
| | Bb F * * D <u>'''A'''</u> E * * C# G#
|-
|-
| style="text-align:center;" | (Phrygian with #3 #7)
| style="text-align:center;" | ''(Phrygian with #3 #7)''
| | 6th Meantone[7] #3 #7
| | ''(6th Meantone[7] #3 #7)''
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
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Using ~16/15
'''[[Octatonic_scale|Diminished]] aka Quadgu''' has pergen (P8/4, P5) and a period of ~6/5. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24. The generator can't be ~10/9, because that would change the mode numbers. The Diminished[4x2] scale has only two modes, because the four genchains have only two notes each. The comma is fifthward, thus the 5th is flattened, and the 32/27 minor 3rd is sharpened. Therefore the 300¢ period is narrower than a m3, and must be a vm3. Both Diminished[4x2] modes:
 
'''[[Octatonic_scale|Diminished]] aka Quadgu''' has a quarter-8ve period of ~6/5. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24. Not ~10/9, because that would change the mode numbers. The Diminished[4x2] scale has only two modes, because the four genchains have only two notes each. The comma is fifthward, so the 5th is flattened, and the 32/27 minor 3rd is > 300¢. Therefore the 300¢ period is narrower than a m3, and must be a vm3. Both Diminished[4x2] modes:


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As long as we stick to MOS scales, terms like Meantone[5] or Meantone[6] are fine. But when we alter, add or drop notes, we need to define what something like "#5" means in a pentatonic or hexatonic context.
As long as we stick to MOS scales, terms like Meantone[5] or Meantone[6] are fine. But when we alter, add or drop notes, we need to define what something like "#5" means in a pentatonic or hexatonic context.


If the scale is written using heptatonically using 7 note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone[5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone[5] #5. If discussing scales in the abstract without reference to any note names, one need to specify which type of numbering is being used.
If the scale is written heptatonically using 7 note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone[5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone[5] #5. If discussing scales in the abstract without reference to any note names, one needs to specify which type of numbering is being used.


The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Because the pentatonic fifth is chroma-negative, the fifthward side of the genchain is flat and the fourthwards side is sharp (assuming a fifth &lt; 720¢). Use "+" for fifthwards and "-" for fourthwards.
The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Because the pentatonic fifth is chroma-negative, the fifthward side of the genchain is flat and the fourthward side is sharp (assuming a fifth &lt; 720¢). Use "+" for fifthwards and "-" for fourthwards.


Using J K L M N for note names, and arbitrarily centering the genchain on L, we get this genchain:
Using J K L M N for note names, and arbitrarily centering the genchain on L, we get this genchain:
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![[Color notation/Temperament Names|color name]]
![[Color notation/Temperament Names|color name]]
! | Ls pattern
! | Ls pattern
! | example in C
! | 19-edo example in C
! | genchain
! | 19-edo genchain
|-
|-
| | heptatonic
| | heptatonic
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See below.
See below.


'''Why not just use UDP notation?'''
'''Why not just use Modal UDP notation?'''


One problem with [[Modal_UDP_Notation|UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.
One problem with [[Modal_UDP_Notation|modal UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.


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Furthermore, UDP uses the more mathematical [https://en.wikipedia.org/wiki/Zero-based_numbering zero-based counting] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.
Furthermore, UDP uses the more mathematical [https://en.wikipedia.org/wiki/Zero-based_numbering zero-based counting] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.


=Related links=
=Other Naming Methods=
 
Modal UDP notation is explained here: [[Modal UDP Notation]]


Jake Freivald has his own method of naming modes here:
Jake Freivald has his own method of naming modes, explained here:  


[[Naming_Rank-2_Scales#Jake Freivald method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method]]
[[Naming_Rank-2_Scales#Jake Freivald method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method]]
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